Reminiscence

In a general sense, it's clear what the pure past would need to be.  It is the substance that all the presents -- the former as well as the current -- are embedded in.  It's some sort of pre-existing, a priori whole that doesn't simply contain the accumulation of the past presents up till now, but is like the collection of all possible presents.  Basically, the pure past is Time itself, which is perhaps one reason why Deleuze very rarely mentions the future in his writings.  The trouble is not in stating this general requirement of what the past needs to do to solve the problem of how there can be memory.  The trouble is in making sense of what this pure past might actually be like for us, or, as Deleuze puts it:

The passive syntheses are obviously sub-representative. The question for us, however, is whether or not we can penetrate the passive synthesis of memory; whether we can in some sense live the being in itself of the past in the same way that we live the passive synthesis of habit. The entire past is conserved in itself, but how can we save it for ourselves, how can we penetrate that in-itself without reducing it to the former present that it was, or to the present present in relation to which it is past? How can we save it for ourselves?

I'd put the emphasis here on the "living" of the past.  As in: how could we experience the past in itself, as the past, and not as a former present encased in and represented to us through a current present?  Deleuze's answer to this question turns out to be straightforward.  We experience the past in itself through reminiscence (which is not the same as memory) and the model for this experience is Proust writing about Combray in The Remembrance of Things Past.  So, before we go into trying to make sense of the more technical and abstract job description of what the pure past needs to be like to solve our memory problem, I want to spend a little time thinking about how me might experience this abstraction directly, like Proust apparently did.

... reminiscence. In effect, this designates a passive synthesis, an involuntary memory which differs in kind from any active synthesis associated with voluntary memory. Combray reappears, not as it was or as it could be, but in a splendour which was never lived, like a pure past which finally reveals its double irreducibility to the two presents which it telescopes together: the present that it was, but also the present present which it could be. Former presents may be represented beyond forgetting by active synthesis, in so far as forgetting is empirically overcome. Here, however, it is within Forgetting, as though immemorial, that Combray reappears in the form of a past which was never present: the in-itself of Combray.

The image of Combray is not the objective image of childhood reproduced for our present as if it were a home movie.  What we're getting in reminiscence is more like the past applied to the present, that part of the past that can help us live now.  This still doesn't quite express the idea though, because the present only is what it is now on the basis of its entanglement with the past.  As in involuntary memory, reminscence doesn't start with a clear construction of what the current present is, and then go looking for a past that might be somehow analogous.  Instead, our experience of the the present is colored by our subjective reading of the past.  In other words, the subjective experience of an involuntary memory (the passive synthesis of memory) creates both the objective former present and the objective current present as tied together by a memory.   This latter is the active synthesis of memory.  It probably shouldn't be called "objective", since it's still a subjective experience; in fact, it's we usually just call "remembering".  But this synthesis is an activity of a subject in a present, whereas the passive synthesis is something the operates beneath or before the activity to establish who the subject is now to begin with.  Spontaneous memory functions as an experience without a stable marker in time that actually creates the present and the past in one motion.  This is what makes it a direct experience of a pure past.  

Turns out this is not a bad description of the very first lines of The Remembrance of Things Past.  Here's how Proust describes the experience.

For a long time I used to go to bed early. Sometimes, when I had put out my candle, my eyes would close so quickly that I had not even time to say "I'm going to sleep." And half an hour later the thought that it was time to go to sleep would awaken me; I would try to put away the book which, I imagined, was still in my hands, and to blow out the light; I had been thinking all the time, while I was asleep, of what I had just been reading, but my thoughts had run into a channel of their own, until I myself seemed actually to have become the subject of my book: a church, a quartet, the rivalry between François I and Charles V. This impression would persist for some moments after I was awake; it did not disturb my mind, but it lay like scales upon my eyes and prevented them from registering the fact that the candle was no longer burning. Then it would begin to seem unintelligible, as the thoughts of a former existence must be to a reincarnate spirit; the subject of my book would separate itself from me, leaving me free to choose whether I would form part of it or no; and at the same time my sight would return and I would be astonished to find myself in a state of darkness, pleasant and restful enough for the eyes, and even more, perhaps, for my mind, to which it appeared incomprehensible, without a cause, a matter dark indeed.

I think this extraordinary image of dreaming that you are awake and thinking that you are living the reality of a fiction helps a lot to get our arms around what the experience of a pure a priori past is like.  One thing for sure -- it's going to lead to a significant loss of our stable identity.